Minimizing a multivariable function given restraint

Question

I want to minimize the following function:

$$J(x, y, z) = x^a + y^b + z^c$$

I know I can easily determine the minimum value of $J$ using partial derivative. But I have also the following condition:

$$ x + y + z = D$$

How can I approach now?

Answer 1

Lagrange multipliers...? Define

$$H(x,y,z,\lambda):=x^a+y^b+z^c-\lambda(x+y+z-D)$$

Find conditions for

$$H_x'=H_y'=H_z'=H_\lambda'=0\;\;\ldots$$

Answer 2

This is an easy example of using Lagrange multiplier. If you reformulate your constraint as $C(x,y,z) = x+y+z-D=0$, you can define $L(x,y,z,\lambda) := J(x,y,z) = J(x,y,z)-\lambda \cdot C(x,y,z)$

If you now take the condition $\nabla L=0$ as necessary for your minimum you will fulfill $$\frac{\partial L}{\partial x}=0 \\ \frac{\partial L}{\partial y}=0 \\ \frac{\partial L}{\partial z}=0 \\ $$ Which are required for your minimum and $$ \frac{\partial L}{\partial \lambda}=C(x,y,z)=0 \\ $$ as your constraint.